cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-9 of 9 results.

A375221 Decimal expansion of 6^((3 + z_0.95)/6), where z_0.95 is the 95% quantile of the standard normal distribution (A329281).

Original entry on oeis.org

4, 0, 0, 3, 1, 3, 0, 7, 5, 9, 6, 0, 0, 8, 5, 2, 2, 4, 7, 2, 0, 0, 9, 6, 2, 3, 7, 3, 9, 3, 3, 5, 6, 5, 9, 2, 0, 6, 3, 8, 2, 2, 6, 3, 0, 8, 8, 2, 6, 3, 1, 9, 0, 1, 5, 6, 5, 7, 9, 6, 3, 2, 7, 4, 1, 8, 3, 8, 0, 9, 5, 5, 6, 7, 6, 3, 7, 6, 8, 4, 6, 1, 2, 1, 3, 1, 0, 3, 6, 0, 7
Offset: 1

Views

Author

Hugo Pfoertner, Aug 10 2024

Keywords

Comments

This constant is often used in the service life requirements for critical parts in aviation, e.g., for the rotor disks of the turbines. The constant, rounded to y=4, serves as a safety factor when only one test result is available for the service life of a component with a log-normal service life distribution with an assumed known +-3 sigma scatter of 6. With a confidence level of 95%, it can then be assumed that the +-3 sigma range of the service life distribution does not fall below the demonstrated test life divided by y.

Examples

			4.0031307596008522472009623739335659206382263...

References

  • Sheldon M. Ross, Introduction to Probability and Statistics for Engineers and Scientists, Third Edition, Elsevier Academic Press, 2004. Chapter 7, Parameter Estimation, 7.3 Interval Estimates, page 242.

Links

Crossrefs

Cf. A329281.

Programs

  • Mathematica
    RealDigits[6^((3 + (x /. FindRoot[10*Erfc[x] == 1, {x, 1, 2}, WorkingPrecision -> 120])*Sqrt[2])/6)][[1]] (* Amiram Eldar, Aug 23 2024 *)
  • PARI
    6^((3 + solve(x=1.0, 2.0, erfc(x)- 1/10)*sqrt(2))/6)

A329280 Decimal expansion of the quantile z_0.9 of the standard normal distribution.

Original entry on oeis.org

1, 2, 8, 1, 5, 5, 1, 5, 6, 5, 5, 4, 4, 6, 0, 0, 4, 6, 6, 9, 6, 5, 1, 0, 3, 3, 2, 9, 4, 4, 8, 7, 4, 2, 8, 1, 8, 6, 1, 9, 9, 0, 7, 8, 2, 4, 3, 5, 2, 5, 8, 2, 6, 5, 9, 7, 0, 2, 6, 4, 8, 2, 3, 0, 5, 6, 5, 7, 0, 3, 3, 2, 4, 8, 1, 2, 2, 4, 5, 4, 3, 0, 1, 5, 5, 4, 3, 8, 1, 6, 1
Offset: 1

Views

Author

Jianing Song, Nov 12 2019

Keywords

Comments

z_p is the number z such that Phi(z) = p, where Phi(x) = Integral_{t=-oo..x} (1/sqrt(2*Pi))*exp(-t^2/2)*dt is the cumulative distribution function of the standard normal distribution. This sequence gives z_0.9 (also called the 9th decile or the 90th percentile).
This number can also be denoted as probit(0.9), where probit(p) is the inverse function of Phi(x). See the Wikipedia link below.

Links

Crossrefs

Quantiles of the standard normal distribution: A092678 (z_0.75), this sequence (z_0.9), A329281 (z_0.95), A329282 (z_0.99), A329283 (z_0.995), A329284 (z_0.999), A329285 (z_0.9995), A329286 (z_0.9999), A329287 (z_0.99999), A329363 (z_0.999999).

Programs

  • Mathematica
    RealDigits[Sqrt[2] InverseErfc[9/10], 10, 100][[1]] (* Jean-François Alcover, Sep 26 2020 *)
  • PARI
    default(realprecision, 100); solve(x=0, 5, erfc(x)-2*0.1)*sqrt(2)

Formula

If X ~ N(0,1), then P(X<=1.2815515655...) = 0.9, P(X<=-1.2815515655...) = 0.1.

A329282 Decimal expansion of the quantile z_0.99 of the standard normal distribution.

Original entry on oeis.org

2, 3, 2, 6, 3, 4, 7, 8, 7, 4, 0, 4, 0, 8, 4, 1, 1, 0, 0, 8, 8, 5, 6, 0, 6, 1, 6, 3, 3, 4, 6, 9, 1, 1, 7, 2, 3, 3, 5, 1, 8, 1, 7, 1, 4, 1, 5, 3, 2, 0, 1, 3, 0, 6, 9, 0, 6, 5, 6, 4, 0, 2, 4, 7, 8, 9, 0, 8, 7, 6, 6, 2, 6, 4, 5, 6, 0, 3, 4, 4, 8, 7, 3, 5, 6, 8, 2, 2, 9, 3, 0
Offset: 1

Views

Author

Jianing Song, Nov 12 2019

Keywords

Comments

z_p is the number z such that Phi(z) = p, where Phi(x) = Integral_{t=-oo..x} (1/sqrt(2*Pi))*exp(-t^2/2)*dt is the cumulative distribution function of the standard normal distribution. This sequence gives z_0.99 (also called the 99th percentile).
This number can also be denoted as probit(0.99), where probit(p) is the inverse function of Phi(x). See the Wikipedia link below.

Examples

			If X ~ N(0,1), then P(X<=2.3263478740...) = 0.99, P(X<=-2.3263478740...) = 0.01.

Links

Crossrefs

Quantiles of the standard normal distribution: A092678 (z_0.75), A329280 (z_0.9), A329281 (z_0.95), this sequence (z_0.99), A329283 (z_0.995), A329284 (z_0.999), A329285 (z_0.9995), A329286 (z_0.9999), A329287 (z_0.99999), A329363 (z_0.999999).

Programs

  • PARI
    default(realprecision, 100); solve(x=0, 5, erfc(x)-2*0.01)*sqrt(2)

A329283 Decimal expansion of the quantile z_0.995 of the standard normal distribution.

Original entry on oeis.org

2, 5, 7, 5, 8, 2, 9, 3, 0, 3, 5, 4, 8, 9, 0, 0, 7, 6, 0, 9, 7, 8, 5, 7, 6, 7, 4, 8, 6, 0, 3, 8, 1, 4, 1, 1, 7, 3, 0, 6, 0, 1, 7, 6, 3, 4, 2, 7, 6, 3, 1, 7, 3, 7, 6, 4, 6, 0, 4, 8, 6, 2, 1, 8, 8, 6, 2, 5, 5, 1, 2, 0, 7, 8, 7, 6, 4, 1, 8, 1, 1, 0, 8, 4, 9, 8, 1, 4, 6, 5, 7
Offset: 1

Views

Author

Jianing Song, Nov 12 2019

Keywords

Comments

z_p is the number z such that Phi(z) = p, where Phi(x) = Integral_{t=-oo..x} (1/sqrt(2*Pi))*exp(-t^2/2)*dt is the cumulative distribution function of the standard normal distribution. This sequence gives z_0.995.
This number can also be denoted as probit(0.995), where probit(p) is the inverse function of Phi(x). See the Wikipedia link below.

Examples

			If X ~ N(0,1), then P(X<=2.5758293035...) = 0.995, P(X<=-2.5758293035...) = 0.005.

Links

Crossrefs

Quantiles of the standard normal distribution: A092678 (z_0.75), A329280 (z_0.9), A329281 (z_0.95), A329282 (z_0.99), this sequence (z_0.995), A329284 (z_0.999), A329285 (z_0.9995), A329286 (z_0.9999), A329287 (z_0.99999), A329363 (z_0.999999).

Programs

  • PARI
    default(realprecision, 100); solve(x=0, 5, erfc(x)-2*0.005)*sqrt(2)

A329284 Decimal expansion of the quantile z_0.999 of the standard normal distribution.

Original entry on oeis.org

3, 0, 9, 0, 2, 3, 2, 3, 0, 6, 1, 6, 7, 8, 1, 3, 5, 4, 1, 5, 4, 0, 3, 9, 9, 8, 3, 0, 1, 0, 7, 3, 7, 9, 2, 0, 5, 4, 9, 1, 0, 0, 8, 4, 9, 1, 8, 6, 5, 8, 0, 8, 8, 5, 5, 6, 9, 7, 1, 7, 1, 1, 0, 8, 5, 4, 3, 5, 6, 9, 1, 4, 2, 8, 9, 5, 1, 4, 5, 5, 5, 3, 1, 2, 2, 6, 6, 7, 2, 4, 1
Offset: 1

Views

Author

Jianing Song, Nov 12 2019

Keywords

Comments

z_p is the number z such that Phi(z) = p, where Phi(x) = Integral_{t=-oo..x} (1/sqrt(2*Pi))*exp(-t^2/2)*dt is the cumulative distribution function of the standard normal distribution. This sequence gives z_0.999.
This number can also be denoted as probit(0.999), where probit(p) is the inverse function of Phi(x). See the Wikipedia link below.

Examples

			If X ~ N(0,1), then P(X<=3.0902323061...) = 0.999, P(X<=-3.0902323061...) = 0.001.

Links

Crossrefs

Quantiles of the standard normal distribution: A092678 (z_0.75), A329280 (z_0.9), A329281 (z_0.95), A329282 (z_0.99), A329283 (z_0.995), this sequence (z_0.999), A329285 (z_0.9995), A329286 (z_0.9999), A329287 (z_0.99999), A329363 (z_0.999999).

Programs

  • PARI
    default(realprecision, 100); solve(x=0, 5, erfc(x)-2*0.001)*sqrt(2)

A329285 Decimal expansion of the quantile z_0.9995 of the standard normal distribution.

Original entry on oeis.org

3, 2, 9, 0, 5, 2, 6, 7, 3, 1, 4, 9, 1, 8, 9, 4, 7, 9, 3, 2, 2, 1, 6, 2, 7, 0, 3, 5, 3, 7, 4, 6, 4, 9, 1, 7, 9, 2, 1, 6, 2, 2, 6, 9, 2, 5, 6, 7, 7, 3, 9, 0, 0, 7, 6, 9, 9, 3, 8, 7, 8, 2, 8, 6, 9, 1, 7, 9, 9, 6, 5, 9, 9, 6, 4, 9, 7, 5, 7, 8, 6, 4, 2, 1, 1, 7, 4, 4, 7, 0, 8
Offset: 1

Views

Author

Jianing Song, Nov 12 2019

Keywords

Comments

z_p is the number z such that Phi(z) = p, where Phi(x) = Integral_{t=-oo..x} (1/sqrt(2*Pi))*exp(-t^2/2)*dt is the cumulative distribution function of the standard normal distribution. This sequence gives z_0.9995.
This number can also be denoted as probit(0.9995), where probit(p) is the inverse function of Phi(x). See the Wikipedia link below.

Examples

			If X ~ N(0,1), then P(X<=3.2905267314...) = 0.9995, P(X<=-3.2905267314...) = 0.0005.

Links

Crossrefs

Quantiles of the standard normal distribution: A092678 (z_0.75), A329280 (z_0.9), A329281 (z_0.95), A329282 (z_0.99), A329283 (z_0.995), A329284 (z_0.999), this sequence (z_0.9995), A329286 (z_0.9999), A329287 (z_0.99999), A329363 (z_0.999999).

Programs

  • PARI
    default(realprecision, 100); solve(x=0, 5, erfc(x)-2*0.0005)*sqrt(2)

A329286 Decimal expansion of the quantile z_0.9999 of the standard normal distribution.

Original entry on oeis.org

3, 7, 1, 9, 0, 1, 6, 4, 8, 5, 4, 5, 5, 6, 8, 0, 5, 6, 4, 3, 9, 3, 6, 6, 0, 6, 2, 4, 5, 0, 8, 4, 7, 8, 3, 0, 4, 6, 1, 7, 3, 1, 9, 7, 0, 8, 2, 7, 2, 1, 5, 4, 6, 8, 4, 7, 3, 9, 4, 8, 1, 7, 2, 4, 7, 8, 6, 9, 3, 0, 6, 4, 3, 2, 9, 6, 7, 2, 6, 1, 7, 8, 9, 0, 7, 2, 7, 0, 3, 2, 7
Offset: 1

Views

Author

Jianing Song, Nov 12 2019

Keywords

Comments

z_p is the number z such that Phi(z) = p, where Phi(x) = Integral_{t=-oo..x} (1/sqrt(2*Pi))*exp(-t^2/2)*dt is the cumulative distribution function of the standard normal distribution. This sequence gives z_0.9999.
This number can also be denoted as probit(0.9999), where probit(p) is the inverse function of Phi(x). See the Wikipedia link below.

Examples

			If X ~ N(0,1), then P(X<=3.7190164854...) = 0.9999, P(X<=-3.7190164854...) = 0.0001.

Links

Crossrefs

Quantiles of the standard normal distribution: A092678 (z_0.75), A329280 (z_0.9), A329281 (z_0.95), A329282 (z_0.99), A329283 (z_0.995), A329284 (z_0.999), A329285 (z_0.9995), this sequence (z_0.9999), A329287 (z_0.99999), A329363 (z_0.999999).

Programs

  • PARI
    default(realprecision, 100); solve(x=0, 5, erfc(x)-2*0.0001)*sqrt(2)

A329287 Decimal expansion of the quantile z_0.99999 of the standard normal distribution.

Original entry on oeis.org

4, 2, 6, 4, 8, 9, 0, 7, 9, 3, 9, 2, 2, 8, 2, 4, 6, 2, 8, 4, 9, 8, 5, 2, 4, 6, 9, 8, 9, 0, 6, 3, 4, 4, 6, 2, 9, 3, 5, 6, 0, 5, 3, 2, 2, 2, 6, 9, 5, 4, 9, 0, 7, 2, 6, 2, 0, 1, 0, 5, 0, 8, 0, 6, 2, 8, 6, 0, 3, 6, 8, 9, 7, 0, 4, 0, 3, 7, 9, 5, 5, 1, 5, 6, 3, 3, 7, 3, 4, 1, 4
Offset: 1

Views

Author

Jianing Song, Nov 12 2019

Keywords

Comments

z_p is the number z such that Phi(z) = p, where Phi(x) = Integral_{t=-oo..x} (1/sqrt(2*Pi))*exp(-t^2/2)*dt is the cumulative distribution function of the standard normal distribution. This sequence gives z_0.99999.
This number can also be denoted as probit(0.99999), where probit(p) is the inverse function of Phi(x). See the Wikipedia link below.

Examples

			If X ~ N(0,1), then P(X<=4.2648907939...) = 0.99999, P(X<=-4.2648907939...) = 0.00001.

Links

Crossrefs

Quantiles of the standard normal distribution: A092678 (z_0.75), A329280 (z_0.9), A329281 (z_0.95), A329282 (z_0.99), A329283 (z_0.995), A329284 (z_0.999), A329285 (z_0.9995), A329286 (z_0.9999), this sequence (z_0.99999), A329363 (z_0.999999).

Programs

  • PARI
    default(realprecision, 100); solve(x=0, 5, erfc(x)-2*0.00001)*sqrt(2)

A329363 Decimal expansion of the quantile z_0.999999 of the standard normal distribution.

Original entry on oeis.org

4, 7, 5, 3, 4, 2, 4, 3, 0, 8, 8, 2, 2, 8, 9, 8, 9, 4, 8, 1, 9, 3, 9, 8, 8, 1, 8, 7, 0, 0, 4, 2, 7, 5, 0, 0, 5, 6, 4, 2, 2, 3, 3, 7, 2, 6, 8, 2, 7, 0, 2, 7, 6, 7, 8, 6, 6, 3, 1, 2, 7, 2, 3, 7, 1, 1, 7, 4, 1, 1, 6, 5, 3, 6, 0, 0, 1, 8, 4, 3, 4, 8, 5, 2, 8, 5, 1, 6, 4, 5, 5
Offset: 1

Views

Author

Jianing Song, Nov 12 2019

Keywords

Comments

z_p is the number z such that Phi(z) = p, where Phi(x) = Integral_{t=-oo..x} (1/sqrt(2*Pi))*exp(-t^2/2)*dt is the cumulative distribution function of the standard normal distribution. This sequence gives z_0.999999.
This number can also be denoted as probit(0.999999), where probit(p) is the inverse function of Phi(x). See the Wikipedia link below.

Examples

			If X ~ N(0,1), then P(X<=4.7534243088...) = 0.999999, P(X<=-4.7534243088...) = 0.000001.

Links

Crossrefs

Quantiles of the standard normal distribution: A092678 (z_0.75), A329280 (z_0.9), A329281 (z_0.95), A329282 (z_0.99), A329283 (z_0.995), A329284 (z_0.999), A329285 (z_0.9995), A329286 (z_0.9999), A329287 (z_0.99999), this sequence (z_0.999999).

Programs

  • PARI
    default(realprecision, 100); solve(x=0, 5, erfc(x)-2*0.000001)*sqrt(2)
Showing 1-9 of 9 results.

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